Problem: $ 2.\overline{3} \div 1.\overline{4} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 23.3333...\\ x &= 2.3333...\end{align*} $ $\begin{align*} 9x &= 21 \\ x &= \dfrac{21}{9}\end{align*} $ $\begin{align*} 10y &= 14.4444...\\ y &= 1.4444...\end{align*} $ $\begin{align*} 9y &= 13 \\ y &= \dfrac{13}{9}\end{align*} $ So, the problem becomes: $ \dfrac{21}{9} \div \dfrac{13}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{21}{9} \times \dfrac{9}{13} = {?} $ $ \phantom{\dfrac{21}{9} \times \dfrac{13}{9}} = \dfrac{21 \times 9}{9 \times 13} $ $ \phantom{\dfrac{21}{9} \times \dfrac{13}{9}} = \dfrac{21 \times \cancel{9}} {\cancel{9} \times 13} $ $ \phantom{\dfrac{21}{9} \times \dfrac{13}{9}} = \dfrac{21}{13} $